Spring 2009 Written Assignment 1: Due Tuesday in lecture! No late - - PDF document

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Spring 2009 Written Assignment 1: Due Tuesday in lecture! No late - - PDF document

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Announcements CS 188: Artificial Intelligence Spring 2009 Written Assignment 1: Due Tuesday in lecture! No late days for written assignments Printed copies will be here after class Lecture 6: Adversarial Search Countdown to

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1 CS 188: Artificial Intelligence

Spring 2009

Lecture 6: Adversarial Search 2/5/2009

John DeNero – UC Berkeley Slides adapted from Dan Klein, Stuart Russell or Andrew Moore

Announcements

  • Written Assignment 1:
  • Due Tuesday in lecture!
  • No late days for written assignments
  • Printed copies will be here after class
  • Countdown to math:
  • Markov decision processes are 3 lectures away
  • Project 2:
  • Posted tonight; due Wednesday, 2/18
  • Material from today and next Tuesday
  • Midterm on Thursday, 3/19, at 6pm in 10 Evans

Game Playing

  • Many different kinds of games!
  • Axes:
  • Deterministic or stochastic?
  • One, two or more players?
  • Perfect information (can you see the state)?
  • Want algorithms for calculating a strategy

(policy) which recommends a move in each state

3

Example: Peg Game

Jump each tee and remove it:

  • Leave only one -- you're genius
  • Leave two and you're purty smart
  • Leave three and you're just plain dumb
  • Leave four or mor'n you're an EG-NO-RA-MOOSE

Looks like a search problem:

  • Has a start state, goal test, successor function
  • But the goal cost is not the sum of step costs!
  • Are all of our search algorithms useless here?

Instructions from Cracker Barrel Old Country Store

Deterministic Single-Player

  • Deterministic, single player,

perfect information games:

  • Start state, successor function,

terminal test, utility of terminals

  • Max search:
  • Each node stores a value: the

best outcome it can reach

  • This is the maximal value of its

children (recursive definition)

  • No path sums; utilities at end
  • After search, can pick

move that leads to the best outcome

Purtysmart Plain dumb Genius Ignoramus 4 3 2 1

Properties of Max Search

Purtysmart Plain dumb Genius Ignoramus

  • Terminology: terminal states,

node values, policies

  • Without bounds, need to search

the entire tree to find the max

  • Computes successively tighter

lower bounds on node values

  • With a known upper bound on

utility, can stop when the global max is attained

  • Nodes are max nodes because
  • ne agent is making decisions
  • Caching max values can

speed up computation

4 3 2 1

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2 Uses of a Max Tree

  • Can select a sequence
  • f moves that

maximizes utility

  • Can recover optimally

from bad moves

  • Can compute values for

certain scenarios easily

Purtysmart Plain dumb Genius Ignoramus 4 3 2 1

Adversarial Search

[DEMO: mystery pacman]

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Deterministic Two-Player

  • Deterministic, zero-sum games:
  • tic-tac-toe, chess, checkers
  • One player maximizes result
  • The other minimizes result
  • Minimax search:
  • A state-space search tree
  • Players alternate
  • Each layer, or ply, consists of a

round of moves

  • Choose move to position with

highest minimax value: best achievable utility against a rational adversary

8 2 5 6 max min

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Tic-tac-toe Game Tree

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Minimax Example

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Minimax Search

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3 Minimax Properties

  • Optimal against a perfect player. Otherwise?
  • Time complexity?
  • O(bm)
  • Space complexity?
  • O(bm)
  • For chess, b

35, m 100

  • Exact solution is completely infeasible
  • Lots of approximations and pruning

10 10 9 100 max min [DEMO: minVsExp]

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Resource Limits

  • Cannot search to leaves
  • Depth-limited search
  • Instead, search a limited depth of tree
  • Replace terminal utilities with an

eval function for non-terminal positions

  • Guarantee of optimal play is gone
  • More plies makes a BIG difference
  • [DEMO: limitedDepth]
  • Example:
  • Suppose we have 100 seconds, can

explore 10K nodes / sec

  • So can check 1M nodes per move
  • reaches about depth 8 – decent

chess program

  • Deep Blue sometimes reached depth

40+ ? ? ? ?

  • 1
  • 2

4 9 4 min min max

  • 2

4

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Evaluation Functions

  • Function which scores non-terminals
  • Ideal function: returns the utility of the position
  • In practice: typically weighted linear sum of features:
  • e.g. f1(s) = (num white queens – num black queens), etc.

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Evaluation for Pacman

[DEMO: thrashing, smart ghosts]

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Why Pacman Starves

  • He knows his score will go

up by eating the dot now

  • He knows his score will go

up just as much by eating the dot later on

  • There are no point-scoring
  • pportunities after eating

the dot

  • Therefore, waiting seems

just as good as eating

Iterative Deepening

Iterative deepening uses DFS as a subroutine:

  • 1. Do a DFS which only searches for paths of

length 1 or less. (DFS gives up on any path of length 2)

  • 2. If “1” failed, do a DFS which only searches paths
  • f length 2 or less.
  • 3. If “2” failed, do a DFS which only searches paths
  • f length 3 or less.

….and so on. This works for single-agent search as well! Why do we want to do this for multiplayer games? … b

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SLIDE 4

4

  • Pruning Example

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  • Pruning
  • General configuration
  • is the best value that

MAX can get at any choice point along the current path

  • If n becomes worse than

, MAX will avoid it, so can stop considering n’s

  • ther children
  • Define

similarly for MIN

Player Opponent Player Opponent

n 22

  • Pruning Pseudocode

v

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  • Pruning Properties
  • This pruning has no effect on final result at the root
  • Values of intermediate nodes might be wrong
  • Good move ordering improves effectiveness of pruning
  • With “perfect ordering”:
  • Time complexity drops to O(bm/2)
  • Doubles solvable depth
  • Full search of, e.g. chess, is still hopeless!
  • This is a simple example of metareasoning

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More Metareasoning Ideas

  • Forward pruning – prune a

node immediately without recursive evaluation

  • Singular extensions – explore
  • nly one action that is clearly

better than others. Can alleviate horizon effects

  • Cutoff test – a decision function

about when to apply evaluation

  • Quiescence search – expand

the tree until positions are reached that are quiescent (i.e., not volatile)

? ? ? ?

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Game Playing State-of-the-Art

  • Checkers: Chinook ended 40-year-reign of human world champion

Marion Tinsley in 1994. Used an endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total

  • f 443,748,401,247 positions. Checkers is now solved!
  • Chess: Deep Blue defeated human world champion Gary Kasparov

in a six-game match in 1997. Deep Blue examined 200 million positions per second, used very sophisticated evaluation and undisclosed methods for extending some lines of search up to 40 ply.

  • Othello: human champions refuse to compete against computers,

which are too good.

  • Go: human champions refuse to compete against computers, which

are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

  • Pacman: unknown

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SLIDE 5

5 GamesCrafters

http://gamescrafters.berkeley.edu/

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